 My Math Forum order of the product of two disjoint cycles is the least common multiple
 User Name Remember Me? Password

 Abstract Algebra Abstract Algebra Math Forum

 January 19th, 2017, 01:51 PM #1 Senior Member   Joined: Jan 2013 From: Italy Posts: 154 Thanks: 7 order of the product of two disjoint cycles is the least common multiple Hi, I have this exercise, but, I have some problems to develop it. Prove that if $\alpha$ represents an r-cycle in $S_n$ and $\beta$ represents an s-cycle, and $\alpha$ and $\beta$ are disjoint, then $o(\alpha \beta) = [r,s]$ (i.e. the least common multiple of $r$ and $s$). END I don't know how to resolve it. So let's see what we have: we have a group $S_n$, and its elements are permutations called as $\alpha, \beta, ...$, permutations can be written in form of cycle, and we take two cycle from the group. $\alpha = (a_1 a_2 ... a_n), \beta = (b_1 b_2 ... b_n)$ by definition two cycles $(a_1 a_2 ... a_n), (b_1 b_2 ... b_n)$ are disjoint if $a_i \ne b_j$ with $i = 1, ..., r; j = 1, ... , s$. For example $(1 2 4), (3 5 6)$ are disjoint, but, $(1 2 4), (3 4 6)$ are not. Also disjoint cycles commute. by definition, the order of $\alpha \beta$ would be the integer $n$ as: $(\alpha \beta)^n = (1) = e$ If I understand, $r,s$, are respectively the number of elements within the cycles $\alpha, \beta$, so it can be possible that $r \ne s$. So, in what way can I prove that it is the least common multiple of r ans s? please, can you give me any help? thanks! January 19th, 2017, 06:55 PM #2 Member   Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 Start by proving that if two permutations are disjoint, then, when raised to a power, the resulting permutations are still disjoint. Then argue that a product of two disjoint permutations can't equal 1 unless they're both equal to 1. Next suppose $(\alpha \beta)^t = 1$. What can you say about $\alpha^t$ and $\beta^t$? Deduce that $r|t$ and $s|t$. Verify that $(\alpha \beta)^{[r,s]} = 1$. Conclude that $o(\alpha \beta)=[r,s]$. Thanks from beesee Tags common, cycles, disjoint, multiple, order, product ### prove the order of 2 disjoint cycles is the lcm

Click on a term to search for related topics.
 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post EvanJ Elementary Math 1 October 25th, 2015 08:06 AM blurgirl Elementary Math 5 July 15th, 2014 07:07 AM Solarmew Applied Math 1 April 12th, 2012 09:01 AM albert90 Real Analysis 1 December 8th, 2009 06:41 AM Yuly Shipilevsky Number Theory 5 March 20th, 2009 11:37 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      